Ma/CS 6b Lecture Notes #8
February 5, 2014
Embeddings of graphs on surfaces
We consider embeddings of graphs on surfaces other than the sphere. The surfaces we
consider are compact 2-manifolds. We use Tg , g 0, to denote the orientable surface of
genus g
Ma/CS 6b Lecture Notes #2
January 10, 2014
Higher connectivity
Recall that a path in a graph is a walk in which all vertex terms are distinct. A
nonnull graph G is connected when any two vertices may be joined by a path in the graph.
If S V (G), the subgr
Ma/CS 6b Problem Set 1
Due 11:59 pm, Thursday, January 17, 2013
[You may have diculty knowing how much to write in doing these problems. Just do the best you can,
and we will make comments. If the statements or denition are not clear, tell me.]
1. (i) Let
Ma/CS 6b Problem Set 3
Due 2:00 pm, Friday, February 1, 2013 Revised Thursday, Jan. 31
1. (i) Show that a graph G all of whose vertices x have even degree has a balanced orientation G. (This
means that indegree(x) = outdegree(x) for every vertex x of the
Ma/CS 6b Problem Set 5
Revised, Feb. 12
Due noon, Saturday, February 16, 2013
(The problems on this set will not be equally weighted; some will be worth more points than others.)
1. Problem 33C in the notes on Coloring and Planarity on the Ma/Cs 6b web pa
Ma/CS 6b Problem Set 2
Due 11:59 pm, Thursday, January 24, 2013
1. (i) Suppose that we allow some edges in a network to have innite capacity. Show that either there
exists a cut of nite capacity or there exists a directed path from s to t all of whose edg
Ma/CS 6b Problem Set 6
Revised, Feb. 21
Due noon, Saturday, February 23, 2013
1. As vertices of K11 , we use the integer cfw_0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 modulo 11. Consider the family of
22 polygons (closed paths) in K11 obtained by taking all transl
Ma/CS 6b Problem Set 7
Due noon, Saturday, March 2, 2013
1. Let e E (G). Explain why the cutsets of Ge are exactly the cutsets of G that do not contain e. Explain
why the cycles of Ge are exactly the cycles of G that do not contain e. (Dont forget to cons
Ma/CS 6b Problem Set 4
Due noon, Saturday, February 9, 2013
1. (i) Let G be a simple graph on n vertices all of whose vertices havee degree at least n/2. Prove that there
is a path of n vertices in G. [Do this directly. Do not use the fact, proved in clas
Ma/CS 6b Problem Set 8
Due noon, Saturday, March 9, 2013
1. [This is Problem 36D of the notes on squared squares. I have tried to help by pointing out what really
needs to be done, namely (i) and (ii) below.]
The cycle space (circulation space) and coboun
Ma/CS 6b Notes January 8, 2014
Parity of permutations
Recall that every permutation of a nite set X has a cycle decomposition. We use
the term rank( ) for n minus the number of cycles of , where |X | = n. For example, the
permutation = (1 4 7 3)(2 5) has
Ma/CS 6b Lecture Notes #6
January 29, 2014
Deletion and contraction
From now on, in this course, we allow loops and parallel edges in graphs. (When
we wish to consider graphs of girth 3, we will say simple graphs.)
Let G be a graph and e E (G). We dene tw
Ma/CS 6b Lecture Notes #7
February 3, 2014
When dealing with graphs with possible loops and parallel edges, we need a slightly
dierent concept than 2-connected, but which is strongly related to it. Let G be a connected
graph. We say G is separable when th
Ma/CS 6b Lecture Notes #9
February 10, 2014
Whitney duality; dual concepts
Many of the proofs in this section are omitted or are sketchy.
We dene the dual graph G of a graph G with respect to a 2-cell embedding of G on
a surface S : The vertices V (G ) ar
Ma/CS 6b Lecture Notes #12
February 24, 2014
Random walks in regular graphs and digraphs
When G is the pentagon, the 5th and 6th powers of the adjacency matrix A were
shown in Notes #10. Since the number of walks in G with any given initial vertex and of
Ma/CS 6b Lecture Notes #13
February 26, 2014
An addressing problem; the number of negative eigenvalues
Another use of linear algebra (or the theory of quadratic forms) is given in this section.
Theorem 1. Let (M ) be any one of the following functions of
Ma/CS 6b Lecture Notes #11
February 19, 2014
Moore graphs of diameter 2
Some interesting graphs have adjacency matrices that satisfy simple matrix equations.
A Moore graph of diameter 2 is a graph of diameter 2 and girth 5. (A graph of
nite girth 6 cannot
Ma/CS 6b Lecture Notes #10
February 12, 2014
Adjacency matrices
Let G be a graph on n vertices. The adjacency matrix of G is the n n matrix A or
A(G) whose rows and columns are indexed by V (G) and where A(x, y ), the entry in row
x and column y , is the
Ma/CS 6b Lecture Notes #5
January 24, 2014
Diracs Theorem
It is not so interesting to ask how many edges are required to force a Hamiltonian
circuit in a graph with n vertices. But we can ask what bound on the minimum degree
will do the job.
Theorem (Dira
Ma/CS 6b Lecture Notes #3
January 15, 2014
Some extremal graph theory
Theorem (Mantel/Turn). A graph on n vertices with no triangles can have at most
a
2
n /4 edges.
Proof: By induction on n (jumping by 2s). The statement of the theorem holds when
n = 0,
Ma/CS 6b Lecture Notes #4
January 22, 2014
Ramseys Theorem
Given intgers a, b 2. the Ramsey number R(a, b) is the least number n so that no
matter how the edges of Kn are colored with two colors, say red and blue, there is either
a set of a vertices so th
Ma/CS 6b Problem Set 9 Revised March 12
Due noon, Sunday, March 17, 2013
If you prefer to not have this due during the study period, you may ask for an extension.
1. Show that a simple graph on n vertices with no triangles and n2 /4 edges is a complete bi